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 JAMP  Vol.8 No.9 , September 2020
An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)
Abstract: Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.
Cite this paper: Bakre, O. , Wusu, A. and Akanbi, M. (2020) An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs). Journal of Applied Mathematics and Physics, 8, 1729-1735. doi: 10.4236/jamp.2020.89130.
References

[1]   Lambert, J.D. (1973) Computational Methods in Ordinary Differential Equations. Academic Press, Cambridge.

[2]   Luke, Y.L., Fair, W. and Wimp, J. (1975) Predictor Corrector Formulas Based on Rational Interpolants. Computers & Mathematics with Applications, 1, 3-12.
https://doi.org/10.1016/0898-1221(75)90003-6

[3]   Fatunla, S.O. (1991) Numerical Methods for IVPs in ODEs. Academic Press, New York.

[4]   Lambert, J.D. and Shaw, B. (1965) On the Numerical Solution of y’ = f(x,y) by a Class of Formulae Based on Rational y’ Approximation. Mathematics of Computation, 19, 456-462.
https://doi.org/10.2307/2003678

[5]   Fatunla, S.O. (1986) Numerical Treatment of Singular Initial Value Problems. An International Journal of Computers and Mathematics with Applications, 128, 1109-1115.
https://doi.org/10.1016/0898-1221(86)90235-X

[6]   Ikhile, M.N.O. (2001) Coefficients for Studying One-Step Rational Schemes for IVPs in ODEs. Computers and Mathematics with Applications, 41, 769-781.
https://doi.org/10.1016/S0898-1221(00)00319-9

[7]   van Niekerk, F.D. (1987) Non-Linear One-Step Methods for Initial Value Problems. Journal of Computational and Applied Mathematics, 13, 367-371.
https://doi.org/10.1016/0898-1221(87)90004-6

[8]   Niekiek, F.D.V. (1988) Rational One Step Methods for Initial Value Problems, Computers & Mathematics with Applications, 16, 1035-1039.
https://doi.org/10.1016/0898-1221(88)90259-3

[9]   Teh, Y.Y. and Yaacob, N. (2013) One-Step Exponential-Rational Methods for the Numerical Solution of First Order Initial Value Problems. Sains Malaysiana, 42, 456-462.

[10]   Fatunla, S.O. (1982) Nonlinear Multistep Methods for Initial Value Problems. An International Journal of Computers and Mathematics with Applications, 8, 231-239.
https://doi.org/10.1016/0898-1221(82)90046-3

[11]   Nkatse, T. and Tshelametse, R. (2015) Analysis of Derivative Free Rational Scheme. MATEMATIKA, 31, 135-142.

[12]   Tasneem, A., Asif, A.S. and Sania, Q. (2018) Development of a Nonlinear Hybrid Numerical Method. Advances in Differential Equations and Control Process, 19, 275-285.
https://doi.org/10.17654/DE019030275

[13]   Ying, T.Y., Omar, Z. and Mansor, K.H. (2014) Modified Exponential-Rational Methods for the Numerical Solution of First Order Initial Value Problems. Sains Malaysiana, 43, 1951-1959.
https://doi.org/10.17576/jsm-2014-4312-18

 
 
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